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# Perfect Polygons

Lesson 4 of 17

## Objective: SWBAT construct regular hexagons, quadrilaterals, and triangles inscribed in a circle. Students will understand how the symmetry of a regular polygon is related to the symmetry of a circle.

#### Lesson Open

*9 min*

**Team Warm-up**

Using the Slide Show, I display the warm-up prompt for the lesson as the bell rings. The prompt asks: Is it ever possible for a quadrilateral to fit inside a circle so that all four vertices lie on the circle? I push students to *explain* their answer (**MP3**). Plausible reasoning is fine at this point, since we have yet to learn formal proof.

The warm-up is an advance organizer for the lesson, which concerns constructions of inscribed polygons. It is also a good check for understanding: do students remember vocabulary like "quadrilateral" and "vertex", and do they see that the circle must intersect the 4 vertices of the polygon. Since the activity follows our Team Warm-up routine, with students sharing their answers and a randomly selected scribe writing the team's answer on the front board, this gives students an opportunity to learn from one another. Students write their answers in their Learning Journals.

As I review the team answers, I look for answers that show students recognize that the question applies to any quadrilateral, not just the one pictured alongside the prompt. At least one team should recognize that a regular quadrilateral is likely to fit inside a circle in the manner described.

**Goal-Setting**

Displaying the Agenda and Learning Targets, I tell the class: When a circle intersects all the vertices of a polygon, the circle is said to "circumscribe the polygon". The polygon is "inscribed in" the circle. As you have guessed, *regular* polygons--those with congruent sides and angles--can always be inscribed in a circle. Why? It is all about symmetry.

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The goal of this activity is not just to teach a construction, but for students to analyze the symmetry of a regular polygon. There are many congruences: not just between the sides and angles of the polygon, but also between the various segments and angles that "come up" in the construction (**MP7**).

This activity also gives students another opportunity to apply the concept that we can only be sure that objects are congruent by superimposing one onto the other (**MP6**). This can be done using tracing paper or a compass--or by folding the construction (**MP5**). The focus is on examining the congruence of the *parts* of the figure (**MP7**).

I plan to assign problems #1 and #3, saving problems #2 and #4 for teams that are working faster than the others.

The lesson uses the Rally Coach format, because I want students to support each other in carrying out the constructions. This activity is a great opportunity to see which students can read the instructions--written in the language and notation of geometry--for understanding. Some students will want to be shown how to perform the constructions. I push them to puzzle through the instructions with the help of their partner (**MP1**). Try something! I will answer specific questions and let the student know if they are going down the right track.

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#### Summarizing Regular Polygons

*10 min*

We summarize regular polygons with the help of the Guided Notes for the lesson.

Although we are not ready to prove that every regular polygon can be inscribed in a circle, later students will see that the rotational symmetry of a regular polygon guarantees that this must be so. In fact, the center of rotation of a regular polygon must be the center of a circle that passes through all the vertices of the polygon. The notes are intended to help students make a connection between regular polygons and circumscribing circles.

More on how I use Guided Notes can be found in my Strategies folder.

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**Team Size-Up**

Displaying the Lesson Close prompt, I ask students to summarize what they learned from the lesson with their team-mates, then select the best answer to write on the board. This activity follows our Team Size-Up routine.

**Homework**

Homework Set 1 problems #12 and 13 review the constructions of inscribed hexagons and quadrilaterals students learned in the lesson. For students who did not get as far, problem #14 (and later #20) introduce the constructions of inscribed triangles and octagons. Students will be able to refer to these problems on the unit quiz.

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- UNIT 1: Models and Constructions
- UNIT 2: Dimension and Structure
- UNIT 3: Congruence and Rigid Motions
- UNIT 4: Triangles and Congruence
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- UNIT 6: Scaling Up- Dilations, Similarity and Proportional Relationships
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- LESSON 1: Previewing Congruence and Rigid Motions
- LESSON 2: Congruence and Coincidence
- LESSON 3: Re-Discovering Symmetry
- LESSON 4: Perfect Polygons
- LESSON 5: Bisector Bonanza
- LESSON 6: The Shortest Segment
- LESSON 7: From Perpendiculars to Parallels
- LESSON 8: Reviewing Congruence
- LESSON 9: Re-Examining Reflections
- LESSON 10: Reconsidering Rotations
- LESSON 11: Taking Apart Translations
- LESSON 12: Visualizing Transformations
- LESSON 13: Reasoning About Rigid Motions
- LESSON 14: Analyzing the Symmetry of a Polygon
- LESSON 15: Reviewing Rigid Motions
- LESSON 16: Rigid Motion and Congruence Unit Quiz
- LESSON 17: Describing Precise Transformations